A finely divided aluminoxane comprising an aluminoxane represented by formula [I] or [II] ##STR1## wherein R denotes a hydrocarbon group having 1 to 10 carbon atoms and m denotes an integer of 2 to 50,an average particle size being 5 to 200 .mu.m and a specific an average particle size being 5 to 200 .mu.m and a specific surface area being 20 to 1000 m.sup.2 /g, a process for producing same, and a catalyst component for polymerization of olefins being said finely divided aluminoxane.